Abstract A half-duplex bidirectional relaying (BR) technique using non-orthogonal multiple access (NOMA) (termed as BR-NOMA) is proposed and investigated in this paper. In BR-NOMA, multiple users in one NOMA group can exchange information simultaneously with multiple users in another NOMA group through a common relay. The optimal information exchanging user set that maximizes the ergodic sum capacity (ESC) of BR-NOMA is investigated. The performance of BR-NOMA is examined in terms of ESC, outage probability (OP), and outage sum capacity (OSC). The analytical expressions of ESC, OP, and OSC are also provided considering the optimal information exchanging user set under both perfect and imperfect successive interference cancellation. Based on the simulation and analytical results, it is demonstrated that BR-NOMA can attain significant capacity gains as compared to half-duplex BR using conventional multiple access.